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St. Petersburg Mathematical Journal

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Lagrangian solutions for the semi-geostrophic shallow water system in physical space with general initial data


Authors: M. Feldman and A. Tudorascu
Original publication: Algebra i Analiz, tom 27 (2015), nomer 3.
Journal: St. Petersburg Math. J. 27 (2016), 547-568
MSC (2010): Primary 76U05
DOI: https://doi.org/10.1090/spmj/1403
Published electronically: March 30, 2016
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Abstract: In order to accommodate general initial data, an appropriately relaxed notion of renormalized Lagrangian solutions for the Semi-Geostrophic Shallow Water system in physical space is introduced. This is shown to be consistent with previous notions, generalizing them. A weak stability result is obtained first, followed by a general existence result whose proof employs the said stability and approximating solutions with regular initial data. The renormalization property ensures the return from physical to dual space and ultimately enables us to achieve the desired results.


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Additional Information

M. Feldman
Affiliation: Department of Mathematics, University of Wisconsin–Madison, Madison, Wisconsin 53706
Email: feldman@math.wic.edu

A. Tudorascu
Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506
Email: adriant@math.wvu.edu

DOI: https://doi.org/10.1090/spmj/1403
Keywords: Semi-Geostrophic Shallow Water system, flows of maps, optimal mass transport, Wasserstein metric, optimal maps, absolutely continuous curves
Received by editor(s): November 25, 2014
Published electronically: March 30, 2016
Additional Notes: The authors would like to thank M. Cullen for his valuable suggestions and comments. The work of Mikhail Feldman was supported in part by the National Science Foundation under Grant DMS-1101260, and by the Simons Foundation under the Simons Fellows program. This work was partially supported by a grant from the Simons Foundation (#246063 to Adrian Tudorascu)
Dedicated: Dedicated to Nina Nikolaevna Ural’tseva on her 80th birthday
Article copyright: © Copyright 2016 American Mathematical Society

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